By Ji L.

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**Additional info for 2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four**

**Example text**

Thus c⊗d ∈ D0 for any c ∈ Crig as claimed. Since I is a right adjoint, it preserves limits [M, Ch. V, §5, Th. 1]. If I preserves colimits then D0 is closed under colimits. Thus D0 is closed under kernels and cokernels and is a full abelian subcategory of D. Here is one of the main definitions in the present paper. 2. Let F : C → D be a tensor functor between arbitrary tensor categories C and D and assume that F has a right adjoint I : D → C. We say that the category D is observable over C if the counit for the adjunction εd : F (I(d)) → d is an epimorphism for all d ∈ D.

If we expand the previous relation then it splits into three components αi αj αk = 0. αi αj αk = 0, αi αj αk = 1, i+j+k=0 i+j+k=1 i+j+k=2 If we replace αi by αi ξ j then these relations will hold and so α0 + α1 ξ + α2 ξ 2 and α0 + α1 ξ 2 + α2 ξ are also central units of finite order. So α0 + α1 ξ + α2 ξ 2 = ξ i g1 and α0 + α1 ξ 2 + α2 ξ = ξ j g2 for some g1 , g2 ∈ G and some integers i, j. Hence (1 − ξ)(α1 + (1 + ξ)α2 ) = g − ξ i g1 . Since 1 − ξ is a nonunit in Z[ξ] this forces g = g1 . Similarly g = g2 .

Introduction Let A be an associative algebra over a commutative ring R with 1. We say that A is graded by a finite abelian group H if A = h∈H Ah where each Ah is an R-submodule of A and Ah1 Ah2 ⊂ Ah1 h2 for all h1 , h2 ∈ H. Our main concern in this paper will be the determination of all possible gradings by abelian groups on integral group rings, that is, when A = ZG and H is abelian. This paper continues our research on gradings in groups rings starter in [BP]. 2. Abelian groups The main significance of this section is to show that if G is a finite abelian group and ZG is graded by C2 , the cyclic group of order 2, then all elements of G are homogeneous.

### 2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four by Ji L.

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Categories: Symmetry And Group